In this section we provide an upper bound to \(EGZ(3,\mathbb {F}_{q}^{n},2)\)  with \(q=3^k\)  (in this section we always consider \(q\)  to be of that form). Even though this is a very special case, providing upper bounds seems to be a hard task. The approach used here is the so-called Slice Rank method, introduced by Terence Tao in [1]} and revisited by Tao and Sawin in [2]} (see also [3]} for a discussion on the method) in order to generalize the polynomial approach introduced in [4]} and in [5]}.
Our application of the method is somehow reminiscent of works on the classical Erdős-Ginzburg-Ziv constants of Fox and Sauermann [6]} and Naslund [7]}.
